
Define the \(\mathbb {C}^\ast \) -invariant subset \(U^{\prime }=\lbrace v^{\prime }\in V^{\prime }:z_1(v^{\prime })=...=z_n(v^{\prime })=0\) . Define the \(\mathbb {C}^\ast \) -equivariant maps \(\iota :U^{\prime }\rightarrow U,\jmath :V^{\prime }\rightarrow V,\Phi ^{\prime }:U^{\prime }\rightarrow V^{\prime },\alpha :V^{\prime }\rightarrow U,\beta :V^{\prime }\rightarrow \mathbb {C}^n\)  by \(\iota :=\check{\alpha }\circ \jmath ^{\prime }|_U^{\prime }\) , \(\jmath :=\check{\jmath }\circ \jmath ^{\prime }\) , \(\Phi ^{\prime }={\rm Id}_{U^{\prime }}\) , \(\alpha =\check{\alpha }\circ \jmath ^{\prime }\)  and \(\beta =(z_1,...,n_n)\) . As \(\jmath ^{\prime }:V^{\prime }\rightarrow \check{V}\) 
is an étale open neighborhood of \(\check{v}\)  in \(\check{V}\)  , there exists \(u^{\prime }\in V^{\prime }\)  with \(\jmath ^{\prime }(u^{\prime })=\check{v}\) ,
and \(z_b(u^{\prime }=0\)  for \(b=1,...,n\)  as \(\check{z}_b(\check{v})=0\) , so \(u^{\prime }\in U\) 
with \(\iota (u^{\prime })= \check{\alpha }\circ \jmath ^{\prime }(u^{\prime }) = \check{\alpha }(\check{v}) = i(x)\) . Also \(\iota ,\jmath ,\alpha \times \beta \)  are étale as \(\check{alpha}\times \check{\beta },\check{\jmath },\jmath ^{\prime }\)  are. The same computations as in the proof of [1]} show that in a neighborhood of \(u^{\prime }\) , which can be taken to be \(\mathbb {C}^\ast \) -invariant, we have \(\Phi \circ \iota =\jmath \circ \Phi \) , so this making \(u^{\prime }\)  smaller and still \(\mathbb {C}^\ast \) -invariant it holds on \(U^{\prime }\) . The
equations \(\alpha \circ \Phi ^{\prime }=\iota ,\beta \circ \Phi ^{\prime }=0\)  are immediate, and \(g\circ \jmath = f\circ \alpha +(2\sum _{b=1}^rz_{2b-1}z_{2b}+\sum _{b=2r+1}^nz_b^2)\circ \beta \)  follows from (REF ). \(\Box \)
